Distributed control of non-linear coupled systems with a single output

ABSTRACT

The present invention encompasses a control system and method for systems of producing and consuming units. The method of the invention includes the steps of setting each producing unit to have an output responsive to an analog signal representative of a market price, and connecting each producing unit to a marketwire, with the changes in the analog signal on the marketwire representing changes in the market price resulting from inputs from the consuming units and the output response of each producing unit.

FIELD OF THE INVENTION

The present invention relates to a method for controlling non-linearcoupled systems with a single output. More particularly, the presentinvention relates to control and coordination of force actuators thatact on a single object without use of a central controller.

BACKGROUND OF THE INVENTION

Market based control requires allocating a task (e.g. physicalactuation) among a large number of producers, with each producer biddingfor part of the task. The task is determined by consumer agents in thesystem (higher level controllers or external requirements). Eachproducer has a supply curve reflecting the actuation or control producedas a function of price and each consumer has a demand curve indicatingthe actuation needed as a function of price. The equilibrium price isdetermined by the price at which aggregate demand and aggregate supplyare equal. The price in turn determines what each individual actuatorproduces and individual consumer uses such that the total actuationequals the demanded actuation. As the task changes, differentcombinations of producers combine to collectively accomplish the task.Such a market is robust against failure of individual agents and changesin tasks, while requiring communication of only one quantity, namely theprice, in order to coordinate the actions of an arbitrary and even timevarying number of producers. Advantageously, such a price based marketsystem naturally provides a Pareto optimal solution that is near optimalallocation within a degenerate array, even though is the fulloptimization problem NP-complete. Furthermore, reconciling conflictinggoals is readily accomplished by having each individual actuator orconsumer individually weight the conflicting goals. The market producesa group resolution between conflicting goals.

Unfortunately, presently available implementation schemes for marketbased control are impractical and scale poorly for large numbers ofproducers or consumers, especially when systems requiring real timeactuation are considered. One could imagine, for example, connecting theproducers and consumers using a bus (e.g. CAN bus, SPI, I²C or Ethernetbus) to communicate pricing information. For example, a 10 Mbit Ethernetsystem could be used to control 10³ nodes of producers and consumers.Even using readily available off-the-shelf hardware, such a marketcontrol system is quite expensive, with per connection node costs ofabout $10 US dollars, for a total cost on the order of $10,000 USdollars. Moreover, due to the long packet header required, the bus wouldtake on the order of 1-10 μsec to transmit a 10 bit number to each node,and then the same time to transmit the results back to each agent forone cycle of the market equilibration loop. If there are such 10³ nodes,this would take on the order 2-20 msecs in the best case without packetcollisions. Other widely available buses (e.g. CAN or SPI) would take atleast an order of magnitude longer. The problem is even worse ifmultiple markets for competing allocations are required. For example, inobject motion control both torques and forces must be allocated eventhough the requirements for each may conflict. If one had 10 markets tocompute the supply and demand curves of competing allocations, eitherone would require 10 buses or the process would take on the order ofseconds. There also is the issue of synchronizing the processors if asynchronous communication scheme is used.

In contrast to conventional digital control schemes, the presentinvention uses a high speed analog system minimally requiring only asingle line (i.e. a “market wire”) interconnecting multiple producersand consumers in a given market. Analog electronic versions of marketsallocate tasks such as actuation or control in a multi-producer andconsumer system using correspondences set up between current and thequantity of a commodity and between the voltage and the price. Themarket consists of consumers removing current from a wire and producersadding current to the wire. The price is the voltage on the wire thateventually reaches an equilibrium price. This voltage, analogous to theequilibrium price, determines the current and hence the actuationproduced by each actuator. As in the case of markets, the performance ofthe system is robust against failures and changes in actuators or tasks.

In one version of an analog system in accordance with the presentinvention, each consumer has a demand curve that decreases linearly asthe voltage on the market wire increases and each producer, such as anactuator or controller, draws current from the wire that serves as the‘market’. The conductances determine the slopes of the supply and demandcurves as well as the market voltage. If some producers produce less,for example, actuation, less current is removed from the wire, thevoltage rises causing more actuation from the remaining producers andreduces the demand by the consumers. Conversely, a decrease in demand(smaller conductances) causes less current to be added to the wirecausing the market wire voltage to decrease. Production thereforedecreases, and demand by the other consumers increases.

Implementation of the foregoing control schemes is of particular utilityfor problems requiring a large number of actuators to produce a desiredactuation level. In this case there would be one consumer with a flatdemand curve, i.e., a current source, for which the current(demand) doesnot change with voltage (price). The actuators (producer agents) wouldproduce actuation such that all this current is removed to ground(demand balanced by supply). This high bandwidth, asynchronouscoordination occurs through one wire and can be inexpensive perconnection (only a few chips per node). No explicit computation isrequired to allocate the resources and a near optimal solution isobtained from a possibly degenerate set of equivalent solutions.

Advantageously, like traditional economic markets, analog circuitimplementations of the present invention are robust against changes inactuator function or failures. As the cost per unit functionality ofsensors, actuators, and computers (agents) continues to decrease,control systems comprised of many interconnected elements becomeincreasingly practical. High speed systems having 10³ processors with 10market wires connected with multiplexed A/D's and D/A's or multiple opamp packages operating in real time are economically feasible usingapparatus and methods of the present invention. Such systems could bemuch more responsive to events in their environment and internal statesas well as exhibit robustness against component failure. Such an analogelectronic implementation is distributed, flexible, easily extensible,efficiently uses wires, and reduces the communication load. Analogmarkets that can compute weighted sums of up to 10⁴ spatiallydistributed agents and communicate the resulting sum back to agents inabout 1-10 μsecs are supportable. The complexity of each such node isabout 1-3 op amps per node or one embedded processor chip per node forthe more flexible implementations.

In one preferred embodiment of the present invention, a distributedmarket based analog control system includes multiple producing units,each producing unit having an output responsive to a market price.Production levels are in part determined by needs of multiple consumingunits, each consuming unit also having an input responsive to a marketprice. Communication of pricing information between the producing unitsand the consuming units is mediated by a marketwire connecting multipleproducing units to multiple consuming units. Absolute or relativevoltage level, current level, or frequency of voltage or current levelchanges can all be used to represent price information on themarketwire. For voltage level based pricing schemes, typically voltagesof about 5-10 volts are used. Since noise based voltage fluctuations onthe marketwire are typically less than about 1.0 mV, as much as 10 bitsof precision is available for distinguishing price levels in the system.To simplify construction, reduce cost, and enhance system robustness,substantially identical adjustable circuit elements can be used for bothmultiple producing units and multiple consuming units, with thesubstantially identical voltage adjustable circuit elements beingconnected to the marketwire.

In a particularly preferred embodiment of the present invention, thedistributed market based control system has no central controller forsetting market prices. In the absence of either a central controller ora central timing mechanism, both of which can be expensive, prone tofailure, or introduce substantial delays in price computation anddistribution, the reliability of the system is improved. In operation.,a producing unit having an output responsive to a market price, aconsuming unit having an input responsive to a market price, areconnected to a marketwire connecting the producing unit to the consumingunit with changes in analog electrical characteristics of the marketwirerepresenting market price fluctuations. These analog electricalcharacteristics of the marketwire can be voltage level changes, currentlevel changes, or time or frequency domain changes in electricalproperties. Such a system operates asynchronously, without a centraltiming clock, with the marketwire immediately transmitting changes inprice information on a microsecond scale, with no need for polling orn-way exchanges of information between n number of producers andconsumers.

Advantageously, the present system for sharing pricing informationbetween producers and consumers is not limited to electrical analogsignals. Other analog propagating physical quantities can be used tocompute the market and communicate pricing information. For example, adistributed market based analog control system including multipleproducing units and multiple consuming units can be based onpartitioning or distribution at least in part on non-electricalproperties, including systems based singly or in combination on changesin electromechanical, mechanical, pressure, temperature, or thermalproperties, chemical concentrations, light levels, or any other suitablephysical property that allows for ready addition or subtraction ofmeasurable system properties (e.g. by substitution of easily measurablefluid pressure changes in a closed pipeline system for voltage changesin wire circuits). Other physical characteristics suitable forcommunication of price include pressure within a cavity, magnetic fluxwithin a superconducting loop, or optical energy within a resonantcavity.

As will be appreciated, since not all producers (consumers) are capableof providing a continuous range of actuation (demand) in response to acontinuous range of received prices, some mechanism for handling stepped(or quantized) actuator response functions without introducing unwantedoscillations is needed. More generally, while it may not be possible tobalance supply and demand exactly at each instance of time it can bepossible to balance the time averaged supply and demand. The presentinvention provides such a mechanism by employment of an inventory orbuffer unit to temporarily store excess output or demand and permitmarket equilibration. The inventory unit may be attached, along withproducing units and consuming units, to a single marketwire orincorporated into individual or groups of producing and/or consumingunits. In operation, for example, the inventory unit can injectadditional charge, raise voltage level, or adjust vibrational amplitudesto smoothly equilibrate the market of consumers and producers.

Such inventory units are particularly useful in conjunction with on/offactuators such as valves. Most often instantaneous valve actuationcannot exact balance the instantaneous demand. For those embodiments ofthe invention having multiple valves, each of the multiple valves has avalve controller to control opening and closure of the multiple valves.A marketwire is connected to each valve controller to convey priceinformation by analog fluctuations in electrical characteristics of themarketwire. Valves can be completely open and completely closed incertain embodiments, while valves controllable to partially open orclose are possible in other embodiments. Valves can be used to controlfluid flow (liquid or air), or even used to control radiant energy (e.g.light valves).

Multiple markets can be used to reconcile conflicting resourceallocation, a problem that nearly always occurs in real systems. Thepresent invention provides a smooth robust balance between conflictinggoals. A market wire is established for each resource to be allocated.The actuators (producers) and consumers for each resource participate ina market that establishes a price for each resource. The utility curvefor each agent (consumer or producer) that participates in more than onemarket, represents a weighted combination of each resource. Theoperation of the aggregate response to the various markets produces arobust, continuous, and optimal solution for conflicting resourceallocation. The analogy to traditional economic markets such theallocation between apples and oranges, for example, is that eachindividuals utility curve expressing their preference for apples versesoranges at a given prices, determines not only how many apples andoranges an individual receives, but also how conflicting resourcedemands for apple and orange production are resolved.

In other embodiments of the invention, a distributed market basedcontrol assembly can be used in conjunction with fixed or movablestructures. Typically multiple actuators are attached to the structure,with each of the multiple actuators having an actuator controller tocontrol actuator applied force. Sensors are used for measuring structuremovement, and a marketwire is connected to each actuator controller toconvey price information to the actuator controllers by analogfluctuations in electrical characteristics of the marketwire. Actuatorscan be used to stabilize a fixed structure against movement, oralternatively can be used to control movement of movable structures fromdefined first positions to second positions (e.g. moving a robotic armso its tip moves from point A to point B).

More generally, the present invention encompasses a control method fornon-linear coupled systems of producing units having a single consumeroutput. The method of the invention includes the steps of setting eachproducing unit to have an output responsive to an analog signalrepresentative of a market price, and connecting each producing unit toa marketwire, with the changes in the analog signal on the marketwirerepresenting changes in the market price and output response of eachproducing unit.

Additional functions, objects, advantages, and features of the presentinvention will become apparent from consideration of the followingdescription and drawings of preferred embodiments.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 schematically illustrates a market controlled system havingproducers and consumers linked by a single marketwire;

FIG. 2 illustrates a conceptual relationship between idealized economicsupply and demand (quantity and price) curves and I-V curves for analogcircuits;

FIG. 3 illustrates a resistive analog circuit having producers andconsumers connected by a marketwire;

FIG. 4 illustrates a transistor based analog circuit having producersand consumers connected by a marketwire;

FIG. 5 illustrates a marketwire based control circuit having embeddedprocessors with A/D and D/A converters;

FIG. 6 illustrates an analog circuit producer employing a light emittingdiode as an actuator whose intensity is set by marketwire based control(any other actuator may;

FIG. 7 illustrates an alternative analog circuit for enabling marketwirebased control of a light emitting diode as an actuator;

FIG. 8 schematically illustrates marketwire circuit that uses currentchanges to represent price fluctuations;

FIG. 9 schematically illustrates a marketwire system that usesnon-electrical communication between consumers and producers to conveyprice information;

FIG. 10 illustrates a supply/demand (quantity/price) curve havingforbidden regions due to quantized control values;

FIG. 11 schematically illustrates use of an inventory circuit tocompensate for any mismatch in demand and supply arising in this casefrom forbidden supply regions due to quantized control values;

FIG. 12 illustrates a marketwire based control circuit and I-V curvehaving hysteresis to control oscillations;

FIG. 13 is a chart illustrating controlled oscillations for threeproducers, with quantized transition supplies, an inventory, and a fixeddemand consumer in a circuit based on that illustrated in FIG. 12;

FIG. 14 illustrates a marketwire based control circuit and I-V curvebased on that illustrated in FIG. 12, with the circuit modified to allowinstant shifts in threshold voltage for improved performance;

FIG. 15 is a chart illustrating controlled oscillations for eightproducers, with quantized transition supplies, an inventory, and a fixeddemand consumer in a circuit based on that illustrated in FIG. 14;

FIG. 16 is an illustrative diagram of a producer/consumer(actuator/sensor) assemblage suitable for applying forces or radiationto external objects;

FIG. 17 is a robotic arm movably controlled by a marketwire system withactuating producers and sensor consumers;

FIG. 18 is a diagram illustrating conflicting resource allocation usingtwo markets in the control of a valve actuator assembly;

FIG. 19 is a circuit used for valve control;

FIG. 20 is a comparison of output of real and simulated of combinedmarkets of force and torque (only force is shown in this figure) for thecontrol of air valve assemblies having three actuators oriented in eachdirection; and

FIG. 21 schematically illustrates embodiments of a market wire includingways of localizing the effects of market supply and demand.

DETAILED DESCRIPTION OF THE INVENTION

As seen in FIG. 1, a distributed market based analog control system 10includes multiple producing units 14, each producing unit having anoutput responsive to a market price. Production levels are in partdetermined by needs of multiple consuming units 12, each consuming unitalso having an input responsive to a market price. As will beappreciated, producing units 14 can be any suitable actuator, includingactuators for applying forces, for generating radiation, for alteringtheir physical properties (e.g. reflectivity changes in liquid crystalassemblies to alter reflection of incident radiation), or forcomputational resource production (e.g. low power digital processorsthat can be selectively brought on and off line as needed). Importantly,for large complex systems, producers and/or consumers may consist oflower or higher level control systems or even other markets. Forexample, a higher level system may be regarded as a consumer for theoutput of a lower level system. The market be used to synchronize theactions of many controllers. In preferred embodiments, actuators can beused to open and close valve assemblies, to provide movement forces forrobotic machinery, move objects within office machines, or to providecountermovement dampening forces to reduce vibrational movement.Consuming units 12 can include sensor systems for control of theforegoing actuator systems, or can themselves be systems for consumingphysical, electrical, or computational resources produced by theproducing units 14.

Communication of pricing information between the producing units and theconsuming units is mediated by a marketwire or groups of marketwires 16connecting multiple producing units 14 to multiple consuming units 12.Absolute or relative voltage level, current level, or frequency ofvoltage or current level changes can all be used to represent priceinformation on the marketwire. In addition, non-electrical signals, suchas may be represented by vibrational modes (e.g. acoustic transmission),radio wave links with amplitude and frequency encoding, heat, pressure,or optical links (e.g. pairwise infrared diode or laser links) betweenproducer and consumer units. For voltage level based pricing schemes,typically voltages of about 5 volts are used. Since noise based voltagefluctuations on the marketwire are typically less than about 1.0 mV andcan be as low as 10 μV within typical bandwidths used in thisapplication, 10 bits or more of precision are available fordistinguishing price levels in the system.

To provide better understanding of the theoretical underpinning of thepresent invention, FIG. 2 is a graphic 20 that illustrates thecorrespondence between traditional price/quantity (i.e. supply/demand)curves 22 and a corresponding analog circuit system with characteristicI-V curves 26. A merged curve explicitly showing the correspondence isshown as graphic 60 (supply/demand curve 60). In graphic 20, curves 22respectively illustrate curves for two producers 30 and two consumers32. An analogous circuit version is illustrated by curves 26, with I-Vcurves 40, produced by two analog circuits 50, matching curves 30, whileI-V curves 42 produced by two sets of analog circuits 52 match curves32. Respective summations of the individual supply and demand curvesproduce aggregate supply and demand curves that determine theequilibrium price Price* or voltage V*, The equilibrium price or voltagethen determines the quantity or current produced or consumed by eachindividual producer and consumer in the market (or analogously, in theanalog circuit). If producers or consumers leave the market or changetheir characteristics (e.g. actuating force available), the market (oranalog circuit) automatically compensates to ensure that supply anddemand (or current/voltage) balance.

FIG. 3 illustrates a simple implementation of an analog market circuit60 having consumers 66, actuating producers 68, connecting marketwire64, and voltage source 62. Each consumer 66 has a demand curve thatdecreases linearly as the voltage on the marketwire 64 increases, andeach producer 68 draws current from the wire that serves as theequivalent of a market. The conductances determine the slopes of thesupply and demand curves as well as the marketwire 64 voltage level. Ifsome producers produce less actuation, less current is removed from thewire, the voltage rises and consequently more actuation is produced bythe remaining producers to reduce the demand from the consumers.Conversely, a decrease in demand (smaller conductances) causes lesscurrent to be added to the wire, consequently resulting less currentadded to the wire (decreasing marketwire 64 voltage). Productiontherefore decreases, and demand by other consumers increases.

While this simple circuit of FIG. 3 is functional, in practice, variouscircuit elements can be provided to account for differences betweenactuators or circuit elements. For example, even seemingly identicalactuators may provide much more actuation than others, particularly ifactuators are discrete or have differing activation (or inactivation)thresholds. Since the lowest threshold actuators would be expected toperform the bulk of the actuation, adjustment of the individual supplyand demand curves is important to ensure equitable distribution. Forexample, this adjustment can be done electronically simply by makingeach variable conductance a transistor for consumers 76 and producers 78connected to marketwire 74 (and voltage source 72) as in circuit 70 ofFIG. 4. The voltages applied to the transistor gates adjust the supplyand demand. The circuit 70 has the desirable feature that it saturatesat high currents, thereby naturally limiting the actuation of eachactuator which corresponds to the current.

More generally, one can create any arbitrary supply and demand curveusing an embedded microprocessors 90 for each consumer 86 and producer88 connected via marketwire 84 as shown in circuit 80 in FIG. 5. Eachprocessor 91 and 92 can control a number of sensors or actuators andparticipate in a number of markets (each of which would have anadditional marketwire, not shown) simultaneously. Also indicated inblock form is the connection between the market, actuation and sensingof the market price. The market price voltage is measured by the A/D andis used to determine the desired actuation or goals. Once this actuation(goal) is determined via the supply (demand) curve, the current isproduced by varying the D/A. The op-amps 93, 95, 94 and 96 buffer themarket from digital signals in the processors. This partiallyanalog/partially digital implementation is the most general and flexiblein that all kinds of market behaviors and temporal variations can beprogrammed into the processor. This flexibility is obtained at thepossible greater cost of a processor and complexity in that eachprocessor must be connected to a bus for programming in the event fixedpre-programming is not sufficient.

Alternatively, an all analog solution for an example case of an LEDproducer using op amps is presented in FIG. 6. This all analog circuit100 for a single producer demonstrates both the reading of the marketvoltage, driving an actuator, and sourcing the appropriate current ontoa market wire 104. The actuator (producer) is an LED 102 that producesradiant energy in this example, but could be any other actuatorincluding force based or discrete actuators. The current removed fromthe wire, I₀=I_(ph)(R₁/R₂), is proportional to the actuation of the LED102 as measured by the phototransistor circuit 106. Any number of theseproducers can be connected up with consumers as shown in previousfigures to yield a market allocation of light. In this case, the marketincludes the actuation as part of the closed loop. Hence, if theactuator fails or ages, the results are compensated by feedback with therest of the market. In this implementation, the speed of the marketadjustment limited by the speed of the actuator.

An alternative pure analog circuit 110 implementation where the marketspeed is not limited by the actuator, shown in FIG. 7. In thisimplementation an analog model of the actuation 116 is used to determinethe response to the voltage on marketwire 114 rather than the actualactuator 117. Because most actuators exhibit complex nonlinear behaviorthat is usually difficult to model in analog circuits, the actuation islinearized using a feedback circuit 115. connected to a V-to-I converter118. The linearized behavior is modeled using saturated analog circuit116 providing input to a linear op-amp circuit 112 that saturates atlarge voltages as does the feedback linearized version of the actuator115. The voltage output of the analog model of the linearized actuation116 is converted to a producer current using the V-to-I converter 118.The analog model of the feedback linearized system 116 is used tocompute the market equilibrium on a rapid time scale. The actuator thenfollows the market voltage on its own time scale to deliver thenecessary actuation corresponding to the market voltage. This circuitwill compute the market equilibrium faster without excessive actuatorswitching.

Using the foregoing analog circuits such as illustrated in FIGS. 5through 7, the aggregate demand and supply of multiple circuits can berapidly determined. If C is the capacitance of the market wire andG_(tot) is the combined total parallel/series conductance of theconsumers and producers, then the time constant is roughly C/G_(tot).For typical lines, C≈1 pF/cm and G_(tot)≈10⁻⁵ Ω⁻¹ and the time constantis 0.1 nsec/cm of market wire. Thus, changes in the actuation arequickly transmitted down the market wire even for long market wires (<10m). If op amps or A/D's and D/A's are used, then their response timedominates the market equilibration times. The slew time of op amps is onthe order of 1 μsec and typical A/D conversion times are on the order of1-10 μsec. Thus, the market computational speed is on the order of 1-10μsecs. The numbers of agents participating in the market are determinedby the minimum measurable currents and the current dynamic range. Ifeach agent adds or subtracts a current roughly equal to 10 times theinput bias current of a reasonable amplifier or 1 nA and the maximumrange of currents is on the order of 1 mA, one might expect that up to10⁶ agents could be connected in an operational market. The ability todetermine the market price is more limited. The noise from powersupplies, Johnson noise, and RF pickup could be as large as 0.5 mVwithout special precautions while the dynamic range is on the order of5V. Including the errors of A/D's, 10 bits is routinely achievable,although up to 16 bits may be achievable at additional cost andcomplexity. The analog market is asynchronous in that there is no needfor coordination among the agents. In fact, asynchrony is desired toallow smoothing of aggregate behaviors. In addition, the agents can beconnected and disconnected without the need for knowledge of the networkstructure. Thus, from these limits about 10³-10⁴ devices could beconnected in a functioning market system using the circuits shown.

In other embodiments of the present invention, the marketwire functionof aggregating supply and demand and communicating price can beaccomplished by separate wires. For example, FIG. 8 illustrates acircuit 130 modeling a single commodity market with consumers 136 andproducers 138 which are aggregated on separate wires 141 and 142respectively. The difference between aggregate supply and demand iscomputed using op amps 137, 139 and 140 and converted to a voltagesignal impressed on a marketwire 134 that transfers information back tothe consumers 136 and producers 138. In operation, consumers add currentto a consumer wire attached to a summing amplifier 137. Similarly,producers 138 add current to a producer wire connected to anothersumming amplifier 139. The respective two current are subtracted in asumming amplifier 140, and output current on the marketwire 134 isrepresentative of commodity price. In this example, the aggregation ofsupply and demand are computed on separate wires from those thatcommunicate price information. Also, as those skilled in the art willappreciate, in these more centralized market schemes permitimplementation of more complex market pricing mechanisms such asauctions.

There are various possible embodiments of the analog market wire asshown in FIG. 21. In general each analog market wire consists aelectrical conductor to transmit voltage and accumulate charge,.Additional markets would require additional such conductors 141, 144,147, and 163. In order for the voltage signal to be common and sharedamong multiple agents, there is an implied additional wire, 142, 143,145, and 162. The second wire is need to transmit the common groundpotential reference for the market voltage. Only one such wire is neededeven if there are numerous markets and market wires. These marketconductors may be simple wires such as twisted pairs 164, shieldedcables 165 or be fabricated as 1d 147 or 2d 148 metal traces on printedcircuit board substrate 146 as in printed circuit boards 166. Metaltrace or traces 147 or 148 form the market wire while the groundpotential is provided by a ground plan. The latter is highly desirabledue to the low cost and ability to fabricate many markets connected tothe agents. In general, shielding is highly desired in order to isolatethe market from extraneous signals that would adversely affect themarket voltage. The market wire conductor may also be augmented bycapacitors or posses distributed capacitance 164 that help damp outsupply and demand fluctuations serving the same purpose as inventories.

An important additional embodiment consists of replacing the market wireby a leaking transmission line 167 with distributed capacitance per unitlength 164 and distributed resistance per unit length along theconductor 161, and a leakage conductance from the market wire to groundper unit length 175 as shown in the lumped circuit representation of atransmission line 167. In this embodiment, the currents I₀, I₁, I₂ etc.(168) due to producers and consumers do not cause a uniform voltage(therefore a price) change throughout the whole market. Instead eachproducer and consumer current changes the price in a localizedneighborhood shown in 169. The localization is a function the values ofdistributed capacitances 164, leakage conductances 175, and resistances161 as is well known from transmission line formulas. This embodiment ofa marketwire can be implemented using discrete resistors and capacitorsor using resistance lines. The trace material 147 may be indium tinoxide rather than copper, the material 145 may be any suitable leakydielectric, and the dimensions of 147 are such to provide the desiredcapacitance 164. The advantage of the leaky transmission line is thateffects of each producer and capacitor are localized. Such a marketwould guarantee that producers and consumers active in the market arespatially distributed along the marketwire. Moreover, if a producer orconsumer malfunctions and attempts to short the market out, the effectswould be localized rather than destroying the entire market. One couldimagine inductors and other distributed or lumped components could beadded to construct a market wire with characteristics beneficial forspecific applications.

In still other embodiments of the present invention, a marketwire cancarry information using other partitionable physical properties orinformation transfer media than that based on changes in electricalcharacteristics. For example, as seen in FIG. 9, a market informationsystem 150 can be based on changes in fluid pressure in variouschambers. The system 150 includes a producer valve 155 for passing apressurized supply fluid from source chamber 158 to the market chamber153. Similarly, consumers 156 can relieve pressure (by removing fluidvia valve 159) from the market chamber 153 to ambient. Changes inpressure of the chamber 153 can be used to transfer market priceinformation between producers and consumers. The pressure resulting fromthe in-flows and out-flows of the market chamber physically determinethe market pressure (price) and communicate the result throughout thechamber with an equilibration time constant given by the ratio of thecapacity of the chamber and the conductance of the chamber. Thisapplication may be useful in coordinating the actions of a number ofpneumatic systems.

In a second example of a non-electrical embodiment of the presentinvention using the same chamber system of FIG. 9, the market isestablished using measurable changes in chemical concentration of H ions(namely, the pH of the market chamber 153) to communicate pricinginformation. In this example the producers 155 admit an acidic solution(low pH solution) from the source chamber 158 into the market chamber153. The consumers admit a basic solution (high pH) from base supplychamber 154 into the market chamber 153. The measured pH of 153represents the market price of the system that is controlled by abalance between the producers and consumers. Such a market can findapplication in distributed reaction systems, biotechnology, incubatingvats, etc where it is important to control many sources of chemicals ina coordinated way.

More generally, any intensive thermodynamic quantity such as pressure,temperature, voltage, chemical potential, magnetic field etc may serveas the market price. For example, sensors distributed in the chambersystem of FIG. 9 can be used to measure fluid temperature changes,changes in acoustic standing wave intensity, or changes in magneticfield intensity. As will be appreciated, the system illustrated in FIG.9 is not limited to pipe-like conduit structures, but in certainembodiments can be modified to form solid connecting beams or structuresto facilitate thermal, acoustic, or magnetic transmission. Whatever theinformation transmission channel employed, the connected consumers andproducers adjust the respective flow, intensity, chemical speciesconcentration, or generalized current of the conjugate extensivequantity to adjust the price. Examples include temperature-energy (heat)flow, pressure-volume flow, chemical potential-concentration flow,voltage-charge flow (current), etc. The utility function for a produceror consumer determines the current of the extensive property passed bythe agent based on the value of the intensive quantity (price). Theaggregate effect of the currents passed by producers and consumerschanges the value of the intensive quantity until a steady state priceemerges.

Whether based on changes in electrical or non-electrical physicalproperties, certain modifications must be made to analog circuits inaccordance with the present invention when supply and demand can not beinstantaneously balanced. This often occurs for example when eitherdiscrete actuators ((on/off or quantized) are producers or discretegoals are consumers. As can be seen with respect to FIG. 10, a supplycurve 170 for a discrete actuator actuation level 174 exhibits discretejumps as a function of price, making the quantity demanded line 172impossible produce instantaneously. In this case the circuit willoscillate about the desired amount such that the time average of thequantity supplied will equal the demand. As seen in FIG. 11, theaveraging process for systems 180 having discrete transition producers182 and consumers 184 can be explicitly controlled by using a capacitor186 to increase the capacitance of the market wire. The capacitor isanalogous to an inventory (or warehouse) that smoothes out fluctuationsin supply and demand in traditional product markets. While theinstantaneous supply and demand are not in balance, the warehouseassures that the time averaged supply and demand remain in balance.

In addition to an inventory circuit analog circuits operating inconjunction with discrete actuators can be implemented with circuitelements imparting predefined hysteresis such as shown in FIG. 12. Asshown in graphic 200, a circuit 202 has hysteretic characteristics(graphic 204). The circuit 200 has an operational amplifier connected ina positive feedback manner to obtain hysteresis, with a transistorconnected as a common base to convert the voltage output to currentoutput. The diode restricts current to positive flow only, and alsoensures that the transistor in circuit 202 operates in its forwardactive region. Connecting such a circuit 202 to three producers withdiscrete actuator transitions and a single consumer demand results intime variation in price and quantity as illustrated in FIG. 13. Thetotal quantity is pulse width modulated to give an average thatsatisfies demand. Note that price rises when producers are notproducing, and fall when producers are producing. In effect, with acircuit 202 and a small number of producer actuators, all the producersturn on for some time, and then turn off for some time, continuallyoscillating to provide a desired average value.

For certain applications, the foregoing behavior (producers all on/alloff) is not desirable since it is sometimes necessary to insure that allactuators or consumers participate in the market so that only a few endup dominating the market (monopolize the market). This more equitableallocation can be implemented using, a circuit 222 shown in Figure toensure that threshold voltage shifts instantly at each turn-on/turn-offtransition, then exponentially decaying back to its original state (seegraphic 224, with step 226 and exponential decay graphic 228illustrated, the transition following formula 227). In operation, for aturn-on transition, the threshold voltage first shifts to the left,making it more difficult for a connected producer to shut off, thengradually decays to the right, increasing the probability of producerturn-off. As would be expected, the opposite occurs for turn-offproducer transitions. The hysteretic circuit causes individual actuatorsto reduce the desire to produce the longer production has continued. Inthis way, the actuation is shared among the producers; no one actuatordoes it all.

The practical utility of the foregoing circuit is seen in FIG. 15, whichillustrates a chart 230 showing time variations for a Producer A and aProducer B randomly selected from a pool of eight producers, aninventory unit, and a consumer with fixed demand using control circuitssuch as discussed in connection with FIG. 14. Note that producers A andB actuation are not in phase, taking turns to produce. This behavioradvantageously prevents large amplitude actuator oscillations and moreevenly distributes actuator workload.

Analog controllers operating under some type of market communicationcould find a useful role in many applications, but is most useful forthose applications requiring many actuators or controllers, and whosecollective behavior is approximately characterized by a weightedsummation of the individual responses. The desirable action of aparticular agent (producer, actuator) must depend on these weightedaverages and location information as opposed to detailed informationabout the state of other particular valves. While these restrictionsappear to limit the utility of the marketwire, actually many problemssatisfy these constraints. For example, as seen in FIG. 16 a system 250in which group of actuators 252 act on objects, such as a collectionheaters heating an object 256, light sources illuminating object 256, orair jets from an array of on/off valves moving a paper sheet 254, forexample, all satisfy these constraints. As seen in FIG. 16, a marketwirecan connect various actuators 260 (that may produce vibrational oracoustical output 275, radiative output 273, or fluid or mechanicalforces 271 directed toward objects) to various sensors 270 (that maydetect vibrational or acoustical input 274, radiative input 272, orfluid or mechanical forces 270, associated with objects).

In addition to heaters, illuminators, valve assemblies for air jets, orother suitable actuator systems, marketwire controllers can be used toboth move and prevent movement of structures. As seen in FIG. 17, anactuator controlled structural assembly 280 can include struts 284interconnected by rotational hinges 286 to form a robotic arm 282movable to various positions. Actuators and sensors (actuators of whichmay be hydraulic assemblies, piezoelectric transducers, or other knownmotion actuators or combination of actuators) tied together by an analogmarketwire and attached to the structural assembly 280 can either causemovement of the struts (generally indicated by large arrows) to a newposition (dotted outline 292) or can counter vibrational movement todampen unwanted motions as needed.

As those skilled in the art will appreciate, any nonlinear-coupledsystems with a single output that has a response that is to first ordera linear sum of the actuations for sufficiently small actuations can bereliably controlled using apparatus in accordance with the presentinvention. If the response R of a system is a function of the vector xof actuations given by R=f(x), by a Taylor expansionR≈f(x₀)+Σ_(i)w_(i)Δx_(i). where w_(i) are the weights related to thefirst derivatives, x₀ is the previous actuation state, and Δx_(i) arethe small changes in the actuation state. Moreover, significant localnon-linearity's can be captured if w_(i)=w_(i) (x_(i)) depends only onthe local state. Hence, the applicability of market based systems isquite general, applying, for example to control of buckling beams orarrays of inverted pendulums, as well as the previously discussed valvearrays, robotic arms, or computational resource problems.

To better understand operation of the present invention, one particularanalog implementation of a market based algorithm for controlling arraysof discrete actuators is hereafter described. In this implementation,motion of a sheet of paper though the use of a very large number ofactuators (open/shut valves for pressurized air jets) must becoordinated to achieve a global goal of precision motion of the sheet ofpaper (see, e.g. FIG. 16 and paper 254). Dynamically, the paper 254moves because the air from each air-jet produces a viscous drag force inthe plane of the paper that is aligned with the direction of theair-jet. The total force on the paper is very nearly the vector sum ofall the forces of the individual air-jets, each of which can produce aconstant force in a given direction. The motion is controlled by turningon and off the individual air-jets located under the paper. Inoperation, the air-jets are constrained not to change state in less timethan, say, 3 milliseconds (3 ms). For example, after an air-jet isturned on, it must wait at least 3 ms before turning off. However, oncethe 3 ms have passed it may turn off at any time. The same is true forthe turning on process. There is also the global constraint that theforce is discrete because each actuator can produce only one fixedamount of force. Position and orientation of the paper must be knownthrough appropriate sensors at all times, and a main controller can beused to combine this information, with a desired paper path, into adesired force. The main control problems are related to determination ofoptimal force allocation, that can take on substantially continuousvalues, as a distribution of large numbers of potentially fallible onand off discrete air-jets.

A typical system of this sort will have a few thousand actuators, with afew hundred actuators lying under the paper at the same time. Typicallyall air-jets will produce the same amount of force, but will point alongdifferent direction in the plane. Assuming a system with motion (of thepaper) along one axis and rotation, each actuator (valved air jet)produces not only a vertical supporting force, but also a torque forcethat is proportional to its perpendicular distance to the center of thepaper. For air-jets distributed randomly along the paper, achieving theoptimal force and torque requires solving an NP-complete problem, whosedifficulty exponentially grows as the number of actuators increases.

The present implementation is based on market principles guiding localdistribution of force, with minimal communication between the air-jetsand the controller, and minimal communication among the air-jetsthemselves. The operation of such a decentralized system can easily becontinuous in time because the switching time between two actuators neednot be a multiple of some fixed time. Because the decentralized systemcan oscillate at a rate much faster than 3 ms even when the amplitude ofoscillations is non-optimal, the root mean square (RMS) error of theintegral will be comparable or even smaller than the equivalent error ofthe centralized controller.

The market principle treats the air-jets as producers and the force andtorque controllers as consumers. The producers and consumers communicatethrough a common voltage (per market) that acts as the price. The priceis determined as a function of aggregate supply and demand, which arefed into the market as currents from the actuators and controllers. Whendemand exceeds supply, the price increases, and either supply increasesor the demand decreases, until equilibrium is reached. Thus, it becomespossible to coordinate a large number of sensors and actuators through asingle market-wire.

The control of many systems often require reconciliation between avariety of constraints. The analog market system can accomplish thisreconciliation through the use of multiple markets. For example, theair-jet system requires at least two markets. Because each actuatorprovides a different ratio of force to torque, the prices of torque andforce must be kept separate. However, the allocation is done in a localfashion, with only the two prices (and their corresponding aggregatecurrent feedback) determining the utility function of each actuatorthereby determining behavior of the system. In fact, there exists noentity (e.g. a central actuator controller) that is aware of thepositions and orientations of the actuators.

FIG. 18 schematically illustrates a suitable air jet loop control system300. System 300 consists of three main components: the producer block302, the consumer block 304, and the warehouse (which can alternativelyand interchangeably be termed a buffer circuit, or an inventory) block306. There is one producer block for each actuator, all of which areconnected in series to the market wires 280 and 281 for force and torquerespectively. There is only one consumer block though, connected to theexternal paper position controller that determines the desired force forpaper movement. If there are multiple controllers simultaneouslycontrolling the paper, there would be multiple consumers and the marketwould synchronize the actions of the multiple controllers. In general, asingle controller will base the desired force on the position of thepaper, so there is an additional feedback loop (not illustrated).

Each producer block 302 is the portion of the circuit that interactswith one actuator (valve). The producer block 302 must observe themarket prices, evaluate using the utility function, and set the state ofthe actuator based on them. Additionally it must feed back its stateinto the force and torque market wires 280 and 281, respectively. Theproducer block consists of three stages: the utility computation, thedecision stage and the feedback to the market. The utility computationstage is the point where each actuator combines the prices from thedifferent markets to obtain a total utility or total price for itsproduct. The utility function any monotonic increasing function of theprices selected to produce desirable behavior of the market. For theair-jet's, the utility beneficially can be a weighted sum of the pricesof force and torque. The weighting factor, for example, may be selectedto be the perpendicular distance to the center of the paper Δycorresponding to the ratio of torque to force that the air-jet produces.The factor Δy multiplies the price of torque, so that the air-jets nearthe center tend to ignore the price of torque, and vice-versa for theair-jets near the edge. Additionally, multiplying by Δy sets the correctrelative sign between torque and force. A backwards facing air-jetshould also multiply the utility by −1, since it produces the exactopposite of a forwards facing air-jet.

For correct scaling between the force and torque terms there must be anadditional constant scaling factor. If the force and torque prices arein the same units of value, the scaling factor should be proportional tothe inverse of the radius of gyration, named R⁻². For a rectangularsheet,

R⁻²=12/(L²+W²) where L is the length and W is the width. Hence, R isabout five inches for an 8.5×11 inch sheet of paper. Summarizing, thefirst stage of the producer block, will compute a utility$U = {\pm \left( {P_{F} + {P_{T}\frac{\Delta\quad y}{R^{2}}}} \right)}$

that characterizes an appropriate tradeoff between the various marketsfor each producer.

The second stage of the producer block consists of the decision stage inwhich each actuator must determine, based on its current utilityfunction, whether it should turn on or off. A first constraint thatmight, beneficially, be taken into account for the air-jets is thatthere must be a delay time, here at least 3 ms, between changes ofstate. Not only does the constraint mean that the decision stage mustignore the utility for 3 ms after a transition, but that in generalthere is a cost to transition between states. Hence the air-jet shouldnot turn on as soon as the utility is positive, or turn off as soon asit is negative, rather it must have some hysteretic threshold. Crossingabove the positive threshold or below the negative threshold, once thefirst 3 ms have passed, should cause the air-jet to change state.

In certain cases there will be actuators with identical utilityfunctions, for example, rows of air-jets at constant Δy. In this case itis desirable to vary slightly the threshold between actuators so thatthe market will react more nearly continuously so that large numbers ofair-jets are not turned on simultaneously. Such collective behavior canlead to excessive oscillation The thresholds can be set to independentlyoscillate in time to avoid having one actuator doing all the work. Inpractice though, many of the air-jets will be waiting for their 3 ms toexpire, so most actuators will get a chance to fire, even if they have ahigh static threshold. The decision stage also controls the actuation ofthe real actuators. Finally, a third stage of a producer, feeds back thestate of the actuator to the market. This is accomplished by applying aforce proportional current to the main force market-wire, 280, and atorque proportional current to the torque market-wire, 281. These ofthese currents are then added by the wire to produce the aggregatesupply for force and torque. In a dynamic setting, each producer musthave an extra sensor that determines if the paper is located above it.As the paper moves, different sets of actuators will be located underthe paper, while the rest produce no force if the jets do not lie underthe sheet of paper. The actuators that are not under the paper mustremove themselves from the market (i.e. not feedback any current) untilafter they are under the paper.

In one preferred implementation of the producer circuit, the market-wirecan be split into two wires: one wire having the market price asvoltage, and another wire with zero voltage that accepts the marketfeedback as current. While this increases the wire count from two tothree within the system (note that any voltage wire requires a groundreference as a implicit or explicitly second required wire), itadvantageously allows market feedback with voltage sources connectedthrough a resistor to the feedback wire, instead of current sources (oneof which needed to be a bipolar voltage controlled current source). Evenmore important is the fact that the utility computation stage, whichnormally requires a voltage multiplier at each site, can be built from asingle resistor ladder.

To implement the utility computation stage of the producer block 302,each actuator is connected to a ladder of resistors such that theresistance between any two actuators is proportional to theirperpendicular distance Δy. The resistor ladder can also be substitutedwith a continuous resistive strip, and each actuator is connected to thenearest part of the strip. All that is required to compute the localutility is to apply voltages at the ends of the strip given by$U = {P_{F} + {P_{T}\frac{r}{R_{2}}}}$

where r is the distance from the each of the ends of the ladder to thecenter of the paper (note that in one case r will be negative). Thisresistive ladder, acting as a voltage divider, substitutes for theutility computation stage at each location and the parts of themarket-wire that carry the price information for both force and torque.

The utility voltage is directly connected to the decision stageconsisting of a voltage comparator with hysteresis. The decision stagecan be built out of an op-amp with circuit 320 as seen in FIG. 19.Without the capacitor, this is a standard comparator with hysteresis,with a threshold voltage of approximately:$V_{T} = {\left( {13\quad V} \right){\frac{R_{T}}{210\quad K\quad\Omega}.}}$

Choosing RT in the range of 8 KΩ to 10 KΩ, provides a threshold valuenear half a volt. However, right after a transition, the capacitor sendsthis threshold voltage to 12 V preventing further transitions until thecapacitor has charged. The value of the capacitor is chosen so that thethreshold approaches 2 V after 3 ms, after which time the actuator isallowed to change state. Because the threshold voltage slowly convergesto its final value even after the 3 ms, recently flipped actuators areless likely to flip again, preventing one actuator from doing all thework.

The last stage of the producer block 302 is the market feedback, whichmust output a current to the market-wire when the actuator is turned on.Supplying the current is simple straightforward when the market-wire issplit into two, since the feedback component is connected to ground(actually, a virtual ground created by an op-amp). A resistor betweenthe power supply and the market-wire provides the necessary current. Theresistor is chosen so that 0.1 mA equals one unit of force, with a signdependent on the direction of the air-jet. The current can be turned onand off by inserting a FET or analog switch between the resistor and themarket-wire.

Minor adjustments must be made in the torque market, since current mustbe proportional to Δy. In a system where the paper can move in thetransverse direction (or when we generalize this method for 2D), eachactuator must be able to calculate Δy. To supply paper positioninformation, a second resistive ladder can be built with a voltagegradient such that zero corresponds to the center of the paper. Eachactuator can use this voltage, with voltage buffer and a suitably chosenresistor to feedback the produced torque into the market.

The demand side of the market in the control loop is particularly simplegiven the existence of a global controller that can determine thedesired force and torque for the paper. The output of the globalcontroller takes on continuous values and is independent of the price(constant demand curve). The consumer block 304 functions by dumping aforce proportional current on the force market-wire, and a torqueproportional current on the torque market-wire where the scale of thecurrent must be the same as the one used by the actuators. The sign ofthe consumer current must be opposite to that of the producers so thatwhen supply equals demand the net current on the wire is zero.Accordingly, the consumer block 304 of the market can be constructedexactly as the feedback circuit for the producer block, with a currentdetermined by the controller.

One final component is required to match the discrepancies between theinstantaneous values of the consumer and actuation supplied by theactuators: a warehouse 306. Functioning as an “inventory unit”, thewarehouse 306 is needed to match the discrete supply with a continuousdemand. One warehouse (capacitor) per market acts as a deposit for theexcess force or torque allowing the time average supply and demand tobalance. Alternatively, each agent could have a capacitor in order tosmooth its individual supply or demand. For voltage based pricingmarkets, the typical warehouse component is a capacitor, whose voltage(price) will be the integral of the difference of demand and supply.

The warehouse block consists simply of a capacitor connected between thevirtual ground of an op-amp, and the output of the op-amp, forming anintegrator. The two prices are fed back into the utility ladder tocomplete the circuit. Since the capacitor voltage should swing betweenthe positive and negative threshold once for every actuator that is notneeded to produce the base force and that has a counterpart in theopposite direction. If the individual thresholds are separated by 1 V,the typical error current is 0.05 mA, and the system has N actuators ineach direction, then a reasonable value for the capacitance would be:$C = {\frac{\left( {{.05}\quad{mA}} \right)\left( {3\quad{ms}}\quad \right)}{\left( {1\quad V} \right)\left( {N/2} \right)} = {\frac{{300\quad{nF}}\quad}{N}.}}$

As will be appreciated, anything having an effective capacitance withinthe same order of magnitude can also be used, including for example, thedistributed capacitance of the market wire itself.

In operation, when demand exceeds supply, the price will begin to rise,and more actuators will turn on, and vice versa. The capacitor will alsokeep track of the net excess of force so that they it be corrected at alatter time. Note that in this market the price can be both positive andnegative. The dual signed price occurs because the market offers bothpositive and negative force. The consumer does not distinguish betweengetting an additional unit of forward force, or one less unit ofbackwards force. A backwards air-jet must behave like a forward air-jet,but based on the negative of the price.

In the simplest case with only one actuator (valve) and one market, themarket will work as a sigma-delta converter. The market will pulse widthmodulate the actuator so that the average supply equals the averagedemand. A more complicated scenario would have still only a force marketbut many actuators. As an example of operation of such a force market,assume that half of the actuators can produce one unit of positiveforce, and the other half can produce one unit of negative force. Ofcourse, they must all have slightly different thresholds, which happensin the real market through their distribution along Δy.

When the one market system is faced with a constant force demand of 2.5units of force, the price will rapidly rise until the forward actuatorwith the lowest threshold is turned on, then slightly slower until thesecond one is turned on. At this point we have the best constant outputthat can be produced. However, the market can do even better by rapidlypulsating between 2 and 3 actuators. With only 2 actuators turned on,the price will continue to rise until the actuator with the third lowestthreshold turns on. After the third actuator turns on, supply willexceed demand and the price will begin to drop. If the price dropsrapidly enough, it will turn on an actuator that produces negativeforce. Once again the force will be 2 units and the price will begin torise, turning on the forth forward actuator. Later the second backwardsactuator will turn on, followed by the fifth forward actuator, and soon. Eventually 3 ms will have passed and actuators will be ready to turnoff. When the price turns negative again, instead of turning on anotherbackwards actuator, it will turn off the first forward actuator,followed eventually by the first backwards actuator. The result will berapid oscillations around the desired force, with a period much smallerthan 3 ms.

Eventually there will be cases when actuators turn on or off at the sametime. Primarily this occurs when the magnitude of the price grows largetriggering an actuator with large threshold, and in the meantime anactuator with low threshold ends its 3 ms waiting period and also flips.This problem can be corrected by forbidding actuator flips when theslope of the utility function is negative. However, the larger the errorin force is, the faster the price changes, so states with excessiveerror are rapidly corrected.

The analysis of a system with both force and torque markets, is somewhatmore complicated but the results are similar. The only caveat is that,given the choice of utility function, the first actuators to turn on tocorrect the torque market are those that provide the highest torque.This is necessary to ensure the stability of the markets, but has theeffect that the torque oscillations will be greater than in othersystems.

A sample system was built with 3 actuators in each direction. The outputof the physical system 352 is compared against an equivalent simulation350 in FIG. 20. The physical system was built with a slightly faster (ascompared to the simulation 350) time constant of about 2 ms, whichexplains the difference in oscillation speed. The physical system alsopresented different stable oscillations for the same input, possiblybecause of the small number of actuators present and the high symmetrywith which they were placed (centered around y=0, with equal spacingbetween them).

Systems built in accordance with the foregoing instructions are known tobe overall stable in the sense that the price, which is the integral ofthe error in force, will always be finite even as time goes to infinity.For one market this is a trivial statement, since no later than 3 msafter crossing the last highest threshold, every actuator will havechanged state to reduce the price. Even if the market overshoots in theother direction, the same condition will apply: the price cannotincrease 3 ms after having crossed the highest threshold. Since thetotal force is bounded, the maximum price, and hence the maximum errorintegrated over all time, is bounded by approximately 3 times the numberof actuators.

With two markets, when the prices of force and torque are high enough,the center actuators will tend to respond to the force market, while theedge actuators will only respond to the torque market. Each group ofactuators cause its market to tend towards time averaged equilibrium, inthe same way described above with a bounded integral of the error.

Another stability criterion that can be asked is whether the market canget trapped in a state of oscillation where all the actuators are on atthe same time, and then all off at the same time. In this undesirablecase, the system is only stable because there are a finite number of.This state can occur in highly symmetrical systems, where actuators havethe same thresholds and the maximum positive torque is equal to themaximum negative torque. In most systems though, a rail to railoscillation will decay to smaller oscillations near the average. Themechanism for this desirable decay that in a non-symmetrical system,there is a torque imbalance that will eventually break the symmetrybetween actuators, preventing them all from turning on at the same time.While one cannot prove that all systems are unconditionally stable froma theoretical standpoint, simulation has shown that this decay generallyhappens for a wide range of systems.

Implementations according to the present invention also exhibit goodscalability properties. From a theoretical standpoint, as the number ofactuators grows, the system oscillates faster, and the RMS of theintegral of the error decreases. Hence, as the number of actuatorsincreases, the system can only perform better. From a practicalstandpoint there are certain issues that must be considered carefully.For example, as the number of actuators increases either the current peractuator decreases, or the total average current in the market wire mustincrease. The high current will require special hardware and consume alot of power, whereas the low current per actuator could result in thesignals from individual actuators being lost in the noise. One way toavoid these problems is to have sub-markets, with a reduced number ofactuators in each, that will produce a fraction of the goal force. Thesub-markets can allocate the force between themselves though anothergreater market. However, for systems of up to a few thousand actuators,a single market should be sufficient to coordinate all the actuators.

Robustness is the greatest advantage achieved by building the system asa decentralized market model. Because there is no central controllerthat must be aware of the positions of each actuator, a single actuatorcan remove itself from the market, without affecting the forceallocation. Robustness is also important because as the paper moves,actuators will slowly enter and exit the market. Thus it is importantthat the system be able to handle these changes gracefully.

So far we have only considered a system with 2 degrees of freedom:torque and force. However, many times it is desirable to move the paperalong the y axis as well. This motion in 2D corresponds to 3 degrees offreedom, and hence 3 independent markets: force along x, force along y,and torque. In a full 2D system, air-jets that push along x will operateas before computing their utility from the prices of force in x, andtorque. Similarly air-jets that push along y will observe the price offorce in y, and the torque. In fact, an actuator that points along anangle θ, can take its force price to beP=P_(xForce)*sin(θ)+P_(yForce)*cos(θ) (which is a simple calculationthat can be done with suitably chosen resistors for each actuator), andthen proceed as normal.

The physical implementation of the 2D system requires 3 wires forcurrent feedback into each of the markets. Additionally, there must betwo resistive strips, one along x and one along y, that will compute theutility function. Finally, there must be another two strips that willcompute Δx and Δy so that the actuators can produce the correct torquefeedback. The stability of the system can be guaranteed in the same wayas with the two market loop. When the prices become too high, the systemuncouples into actuators fixing the x force, actuators fixing the yforce, and actuators fixing the torque. Otherwise the system should workas desired.

As those skilled in the art will appreciate, the error in position canstill diverge. For example, let the desired force be 0.5 units of force,and let the produced force switch between 0 and 1 units of force with aperiod T. Then $\begin{matrix}{{\Delta\quad{V(t)}} \equiv {\int{\Delta\quad{F(t)}{{\mathbb{d}t}/M}}}\quad \propto \frac{T}{8}} \\{{\Delta\quad{X(t)}} \equiv {\int{\Delta\quad{V(t)}{\mathbb{d}t}}} \propto {\frac{T}{8} \cdot t}}\end{matrix}$

where ΔF is the error in force, ΔX(t) is the error in position of thepaper and could be unbounded with time, and ΔV(t) and M is the mass ofthe paper

While the position error diverges as time goes to infinity, the errorcan be reduced by decreasing the period, thus our emphasis on rapidoscillations. However, to eliminate the possibility of divergencecompletely, the market must compensate for the cumulative error inposition. This is accomplished by adding a double integral term to theprice:P=∫ΔF dt+C∫∫ΔF dt dt

where C is some suitably chosen constant.

In practice though, for a system with enough actuators, this error inposition is generally small relative to the physical errors that affectthe paper such as differences in air-jets, unwanted air-currents, and soon. The position error produced by the market can usually be fixed bythe paper controller, along with all the other errors.

As those skilled in the art will appreciate, other variousmodifications, extensions, and changes to the foregoing disclosedembodiments of the present invention are contemplated to be within thescope and spirit of the invention as defined in the following claims.

1. A control method for non-linear coupled systems of producing unitshaving a single consumer output, the method comprising the steps ofsetting each producing unit to have an output responsive to acontinuously provided analog signal representative of a market price,connecting each producing unit to a marketwire carrying the analogsignal, with the changes in the analog signal on the marketwirerepresenting changes in the market price and dependent upon the outputresponse of each producing unit.
 2. A control method of claim 1, whereinthe analog signal is electrical.
 3. The control method of claim 2,wherein the electrical analog signals on the marketwire are representedby voltage levels and changed by varying the voltage levels in themarketwire.
 4. The control method of claim 2, wherein the electricalanalog signals on the marketwire are represented by current levels andchanged by varying the current levels in the marketwire.
 5. A controlmethod for systems of interconnected producing units and consumingunits, the control method comprising the steps of providing producingunits connected to a marketwire, the producing units having an outputresponsive to a market price determinable directly from a signal carriedby the marketwire, providing consuming units connected to themarketwire, the consuming units having an input responsive to the marketprice determinable directly from the signal carried by the marketwire,and equilibrating electrical characteristics on the marketwire based oninput/output from the producing units and the consuming units todetermine the market price.
 6. The control method of claim 5, whereinthe electrical characteristic equilibrated is voltage level in themarketwire.
 7. The control method of claim 5, wherein the electricalcharacteristic equilibrated is current in the marketwire.
 8. A controlmethod for systems of interconnected producing units and consumingunits, the control method comprising the steps of providing producingunits connected to a marketwire, the producing units having an outputresponsive to a market price determinable directly from a signal carriedby the marketwire, providing consuming units connected to themarketwire, the consuming units having an input responsive to the marketprice determinable directly from a signal carried by the marketwire,storing excess electrical energy in an inventory unit as necessary toequilibrate electrical characteristics on the marketwire based oninput/output from the producing units and the consuming units todetermine the market price.
 9. The control method of claim 8, whereinthe electrical characteristic equilibrated is voltage level in themarketwire.
 10. The control method of claim 8, wherein the electricalcharacteristic equilibrated is current in the marketwire.